MANAGEMENT SCIENCE
informs
Vol. 50, No. 9, September 2004, pp. 1178–1192
issn 0025-1909 eissn 1526-5501 04 5009 1178
®
doi 10.1287/mnsc.1030.0163
© 2004 INFORMS
Option Pricing Under a Double Exponential
Jump Diffusion Model
S. G. Kou
Department of IEOR, Columbia University, 312 Mudd Building, New York, New York 10027, sk75@columbia.edu
Hui Wang
Division of Applied Mathematics, Brown University, Box F, Providence, Rhode Island 02912, huiwang@cfm.brown.edu
A
nalytical tractability is one of the challenges faced by many alternative models that try to generalize the
Black-Scholes option pricing model to incorporate more empirical features. The aim of this paper is to
extend the analytical tractability of the Black-Scholes model to alternative models with jumps. We demonstrate that a double exponential jump diffusion model can lead to an analytic approximation for finite-horizon
American options (by extending the Barone-Adesi and Whaley method) and analytical solutions for popular
path-dependent options (such as lookback, barrier, and perpetual American options). Numerical examples
indicate that the formulae are easy to implement, and are accurate.
Key words: contingent claims; high peak; heavy tails; volatility smile; overshoot
History: Accepted by Ravi Jagannathan, finance; received December 3, 2002. This paper was with the authors
3 months for 2 revisions.
1.
Introduction
(d) affine stochastic volatility and affine jump diffusion models; and (e) models based on Lévy processes.
For the background of these alternative models, see,
for example, Hull (2000) and Carr et al. (2003).
Unlike the original Black-Scholes model, although
many alternative models can lead to analytic solutions for European call and put options, it is difficult to do so for path-dependent options, such
as American options, lookback options, and barrier
options. Even numerical methods for these derivatives are not easy. For example, the convergence rates
of binomial trees and Monte Carlo simulation for
path-dependent options are typically much slower
than those for call and put options; for a survey, see,
for example, Boyle et al. (1997).
This paper attempts to extend the analytical
tractability of Black-Scholes analysis for the classical
geometric Brownian motion to alternative models
with jumps. In particular, we demonstrate that a
double exponential jump diffusion model (Kou 2002)
can lead to analytic approximation for finite-horizon
American options by extending the approximation in
Barone-Adesi and Whaley (1987) for the classical geometric Brownian motion model, and analytical solutions for lookback, barrier, and perpetual American
options.
The paper is organized as follows. Section 2 gives a
basic setting of the double exponential jump diffusion
model, and presents intuition on why the analytical
solutions are possible. An analytical approximation
Much research has been conducted to modify the
Black-Scholes model based on Brownian motion and
normal distribution in order to incorporate two
empirical features: (1) The asymmetric leptokurtic
features. In other words, the return distribution is
skewed to the left, and has a higher peak and two
heavier tails than those of the normal distribution.
(2) The volatility smile. More precisely, if the BlackScholes model is correct, then the implied volatility
should be constant, but it is widely recognized that
the implied volatility curve resembles a “smile,”
meaning that it is a convex curve of the strike price.
To incorporate the asymmetric leptokurtic features
in asset pricing, a variety of models have been
proposed, including, among others, (a) chaos theory,
fractal Brownian motion, and stable processes; (b) generalized hyperbolic models, including log t model and
log hyperbolic model; and (c) time-changed Brownian
motions, including log variance gamma model. In a
parallel development, different models are also proposed to incorporate the “volatility smile” in option
pricing. Popular ones are (a) stochastic volatility1
and GARCH models; (b) constant elasticity of variance (CEV) model; (c) normal jump diffusion models;
1
One empirical phenomenon worth mentioning is that the daily
return distribution tends to have more kurtosis than the distribution of monthly returns. As Das and Foresi (1996) point out,
this is consistent with models with jumps, but inconsistent with
stochastic volatility models or other pure diffusion models.
1178
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1179
Management Science 50(9), pp. 1178–1192, © 2004 INFORMS
of finite-time American options is given in §3, and
the analysis of other path-dependent options is conducted in §4. The concluding remarks are given in §5.
All the proofs are given in the appendices.
2.
Background and Intuition
2.1.
The Double Exponential Jump
Diffusion Model
Under the double exponential jump diffusion model,
the dynamics of the asset price St is given by
N t
dSt
Vi − 1 = dt + dW t + d
St−
i=1
where W t is a standard Brownian motion, N t a
Poisson process with rate , and Vi a sequence of
independent identically distributed (i.i.d.) nonnegative random variables such that Y = logV has an
asymmetric double exponential distribution with the
density
fY y = p ·1 e
−1 y
1y≥0 +q ·2 e
2 y
1y<0 1 > 1 2 > 0
where p q ≥ 0, p + q = 1. Here the condition 1 > 1 is
imposed to ensure that the asset price St has finite
expectation. Note that the means of the two exponential distributions are 1/1 and 1/2 , respectively. In
the model, all sources of randomness, N t, W t, and
Y s, are assumed to be independent.
Because of the jumps, the risk-neutral probability
measure is not unique. Following Lucas (1978) and
Naik and Lee (1990), it can be shown (see, e.g.,
Kou 2002) that, by using the rational expectations
argument with a HARA-type utility function for the
representative agent, one can choose a particular riskneutral measure2 P∗ so that the equilibrium price of
an option is given by the expectation under this
risk-neutral measure of the discounted option payoff.
Under this risk-neutral probability measure, the asset
price St still follows a double exponential jump diffusion process:3
∗
N t
∗
dSt
∗ ∗
∗
Vi −1 (1)
= r − dt +dW t+d
St−
i=1
with the return process Xt = logSt/S0 given by
Xt = r − 12 2 − ∗ ∗ t
∗
+ W t +
N ∗ t
i=1
2
3
Yi∗ X0 = 0
(2)
Here W ∗ t is a standard Brownian motion under
P∗ , N ∗ t; t ≥ 0 is a Poisson process with intensity
∗
∗ V ∗ = eY . The log jump sizes Y1∗ Y2∗ · · · still
form a sequence of i.i.d. random variables with
a new double exponential density fY ∗ y ∼ p∗
∗
∗
1∗ e−1 y 1y≥0 +q ∗ 2∗ ey2 1y<0 . The constants p∗ q ∗ ≥ 0,
∗
∗
∗
∗
p + q = 1, > 0, 1 > 1, 2∗ > 0, and
∗ = E∗ V ∗ ! − 1 =
p∗ 1∗
q ∗ 2∗
+
−1
1∗ − 1 2∗ + 1
(3)
all depend on the utility function of the representative
agent. All sources of randomness, N ∗ t, W ∗ t, and
Y ∗ s, are still independent under P∗ .
Since we focus on option pricing in this paper, to
simplify the notation, we shall drop the superscript
∗ in the parameters, i.e., using p, q, 1 , 2 rather
than p∗ , q ∗ , 1∗ , 2∗ . The understanding is that all the
processes and parameters below are under the riskneutral probability measure P∗ .
2.2. Intuition of the Pricing Formulae
Without the jump part, the model simply becomes the
classical geometric Brownian motion model. Pricing
formulae for American options, barrier options, and
lookback options are all well known under the geometric Brownian motion model.4 With the jump part,
however, it becomes very difficult to derive analytical
solutions for these options.
The reason for that is as follows. To price American
options, barrier options, and lookback options for
general jump diffusion processes, it is crucial to study
the first passage times that the process crosses a flat
boundary with a level b. Without loss of generality,
assume b > 0. When a jump diffusion process crosses
the boundary, sometimes it hits the boundary exactly
and sometimes it incurs an “overshoot,” X#b − b,
over the boundary, where #b is the first time that the
process Xt crosses the boundary. See Figure 1 for an
illustration.
The overshoot presents several problems, if one
wants to compute the distribution of the first passage
times analytically. First, one needs the exact distribution of the overshoot, X#b − b; particularly, P X#b −
b = 0! and P X#b − b > x!, x > 0. Second, one needs
to know the dependence structure between the overshoot, X#b − b, and the first passage time #b . Both
difficulties can be resolved under the assumption that
the jump size Y has an exponential type distribution;
see Kou and Wang (2003). Finally, if one wants to
use the reflection principle to study the first passage
times, the dependence structure between the overshoot and the terminal value Xt is also needed
The measure P∗ is called risk neutral because E∗ e−rt St = S0.
For option pricing, the case of the underlying asset having a continuous dividend yield can be easily treated by changing r to
r − in (1).
4
Davydov and Linetsky (2001, 2003) provide analytical solutions
for various path-dependent options under the CEV diffusion
model.
Kou and Wang: Double Exponential Jump Diffusion Model
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Management Science 50(9), pp. 1178–1192, © 2004 INFORMS
Figure 1
A Simulated Sample Path with the Overshoot Problem
If the jump size distribution is one sided, one can
solve the overshoot problems7 by either using renewal
equations or fluctuation identities for Lévy processes;
see, e.g., Avram et al. (2001) and Rogers (2000). However, for two-sided jumps, because of the laddervariable problems, generally speaking the renewal
equations are not available and the fluctuation identities become too complicated for explicit computation; see, e.g., the discussion in Siegmund (1985) and
Rogers (2000).
2.3. Some Notations
The moment-generating function of Xt is given
by E∗ e&Xt ! = expG&t, where the function G· is
defined as
afterward. To the best of our knowledge, this is not
known, even for the double exponential jump diffusion process.
Consequently, we can get analytic approximations
for finite-horizon American options, closed-form solutions for the Laplace transforms of lookback and
barrier options, and closed-form solutions for the perpetual American options,5 under the double exponential jump diffusion model, yet cannot give more
explicit calculations beyond that, as the correlation
between the terminal state Xt and the overshoot
X#b − b is not available.6
It is worth mentioning that the double exponential jump diffusion process is a special case of Lévy
processes with two-sided jumps, whose characteristic
exponent admits the (unique) representation
Gx = x r − 12 2 − + 12 x2 2
p1
q2
+
+
−1 1 − x 2 + x
Lemma 3.1 in Kou and Wang (2003) shows that the
equation Gx = +, ∀+ > 0, has exactly four roots: ,1 + ,
,2 + , −,3 + , and −,4 + , where
0 < ,1 + < 1 < ,2 + < 0 < ,3 + < 2 < ,4 + < To use the Itô formula for jump processes, we also
need the infinitesimal generator of Xt:
V x = 12 2 V x + r − 12 2 − V x
+
V x + y − V x!fY y dy (5)
%& = E ei&X1 !
= exp i'& − 12 A& 2
+
−
(4)
−
ei&y − 1 − i&y1y≤1 )dy where the generating triplet ' A ) is given by
A = 2
)dy = · fY ydy
= p1 e−1 y 1y≥0 dy + q2 e2 y 1y<0 dy
' = + E V 1V ≤1 !
1 − e−1
1 − e−2
−1
−2
= + p
− q
−e
−e
1
2
5
Essentially, to compute the values of perpetual American options,
one only needs to know the Laplace transforms (there is no need
to invert the transforms). Hence, we can get closed-form solutions
for perpetual American options.
6
See, for example, Siegmund (1985), Boyarchenko and Levendorskiı̆
(2002b), and Kyprianou and Pistorius (2003) for some representations (though not explicit calculations) related to the overshoot
problems for general Lévy processes.
3.
Pricing Finite Time Horizon
American Options
Most of call and put options traded in the exchanges
in both the United States and Europe are Americantype options. Therefore, it is of great interest to calculate the prices of American options accurately and
quickly. The price of a finite-horizon American option
is the solution of a finite-horizon free boundary problem. Even within the classical geometric Brownian
motion model, except in the case of the American call
7
For a jump diffusion process with one-sided jumps, the overshoot
problem may not occur if the boundary level is on the opposite
direction of the jumps; e.g., the jump sizes are negative and the
boundary level b > 0. Under this circumstance, explicit solution
may be possible (at least for perpetual American options) even
if the distribution of jump size takes a general form; see, e.g.,
Mordecki (1999).
Kou and Wang: Double Exponential Jump Diffusion Model
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Management Science 50(9), pp. 1178–1192, © 2004 INFORMS
option with no dividend, there is no analytical solution available.8
To price American options under general jump
diffusion models, one may consider numerically
solving the free boundary problems via lattice or differential equation methods; see, e.g., Amin (1993),
Zhang (1997), d’Halluin et al. (2003). Extending the
Barone-Adesi and Whaley (1987) approximation for
the classical geometric Brownian motion model, we
shall consider an alternative approach that takes into
consideration the special structure of the double exponential jump diffusions. One motivation for such an
extension is its simplicity, as it yields an analytic
approximation that only involves the price of a
European option. Our numerical results in Tables 1
and 2 suggest that the approximation error is typically less than 2%, which is less than the typical bidask spread (about 5% to 10%) for American options
in exchanges. Therefore, the approximation can serve
as an easy way to get a quick estimate that is perhaps
accurate enough for many practical situations.
The extension of Barone-Adesi and Whaley’s (1987)
method works nicely for double exponential jump diffusion models mainly because explicit solutions are
available to a class of relevant integro-differential free
boundary problems; see Equations (A3) and (A4) in
Appendix A. We want to point out that there exist
other more elaborate but more accurate approximations (such as Broadie and Detemple 1996, Carr 1998,
and Ju 1998) for geometric Brownian motion models, and whether these algorithms can be effectively
extended to jump diffusion models invites further
investigation.
To simplify notation, we shall focus only on the
finite-horizon American put option without dividends, as the methodology is also valid for the
finite-horizon American call option with dividends.
The analytic approximation involves two quantities:
EuPv t, which denotes the price of a European
put option with initial stock price v and maturity
t, and Pv St ≤ K!, which is the probability that
the stock price at t is below K with initial stock
price v. Both EuPv t and Pv St ≤ K! can be computed fast by using either the closed-form solutions
in Kou (2002) or the Laplace transforms in Kou et al.
(2003).
We need some notations. Let z = 1 − e−rt , ,3 ≡ ,3 r/z ,
,4 ≡ ,4 r/z , C, = ,3 ,4 1 + 2 , D, = 2 1 + ,3 1 + ,4 ,
8
For recent developments of numerical solution and analytic
approximation of finite-horizon American options within the classical geometric Brownian motion model, see, for example, Broadie
and Detemple (1996), Carr (1998), Ju (1998), Geske and Johnson
(1984), McMillan (1986), Tilley (1993), Tsitsiklis and van Roy (1999),
Sullivan (2000), Broadie and Glasserman (1997), Carriére (1996),
Longstaff and Schwartz (2001), Rogers (2002), Haugh and Kogan
(2002), and references therein.
in the notation of Equation (4). Define v0 ≡ v0 t ∈
0 K as the unique solution9 to the equation
C, K − D, v0 + EuPv0 t!
= C, − D, Ke−rt · Pv0 St ≤ K!
(6)
Note that the left-hand side of (6) is a strictly
decreasing function of v0 (because v0 + EuPv0 t =
e−rt E∗ maxSt K S0 = v0 !, and the right-hand
side of (6) is a strictly increasing function of v0
(because C, − D, = ,3 ,4 − 2 1 + ,3 + ,4 < 0. Therefore, v0 can be obtained easily by using, for example,
the bisection method.
Approximation. The price of a finite-horizon
American put option with maturity t and strike K can be
approximated by 2S0 t, where the value function 2 is
given by
EuPv t + Av−,3 + Bv−,4 if v ≥ v0
(7)
2v t =
K − v
if v ≤ v0
with v0 being the unique root of Equation (6) and the two
constants A and B given by
,
A=
v0 3 , K − 1 + ,4 v0 + EuPv0 t!
,4 − ,3 4
+ Ke−rt Pv0 St ≤ K! > 0 (8)
,
B=
v0 4 , K − 1 + ,3 v0 + EuPv0 t!
,3 − ,4 3
+ Ke−rt Pv0 St ≤ K! > 0 (9)
Tables 1 and 2 present numerical results for finitehorizon American put options, corresponding to t =
025 and t = 10 years. The parameters used here are
S0 = 100, p = 06, and r = 005. To save space, the
numerical results for t = 05 and t = 15 are omitted, but they can be obtained from the authors upon
request. We choose this set of maturities because most
of the American options traded in exchanges have
maturities between three months and one year. The
“true” value is calculated by using the enhanced binomial tree method as in Amin (1993) with 1,600 steps
(to ensure that the accuracy is up to about a penny)
and the two-point Richardson extrapolation for the
square-root convergence rate.
In the tables the maximum relative error is only
about 2.6%, while in most cases the relative errors
are below 1%. Note also the approximation tends to
works better for small maturity t; this is because of
Assumption (A2) in the approximation, as will be
explained in Appendix A.1. All the calculations are
conducted on a Pentium 1500 PC. The approximation runs very fast, taking only about 0.04 second to
9
In Appendix A, we give a better upper bound in (A6) for v0 , that
is K > v0 + EuPv0 t.
Kou and Wang: Double Exponential Jump Diffusion Model
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Management Science 50(9), pp. 1178–1192, © 2004 INFORMS
Table 1
Comparison of the Approximation and the True Value for Finite-Horizon American Put Option with t = 025 Year
Parameter values
1
2
“True” value (a)
CPU time
Approx. (b)
CPU time
Abs. error (b) − (a)
Relative error
((b) − (a))/(a) (%)
110
110
110
110
110
110
110
110
110
110
110
110
110
110
110
110
02
02
02
02
02
02
02
02
03
03
03
03
03
03
03
03
3
3
3
3
7
7
7
7
3
3
3
3
7
7
7
7
25
25
50
50
25
25
50
50
25
25
50
50
25
25
50
50
25
50
25
50
25
50
25
50
25
50
25
50
25
50
25
50
1048
1042
1036
1031
1081
1068
1051
1039
1190
1184
1178
1172
1223
1209
1194
1180
4,030
4,029
4,029
4,027
4,030
4,029
4,028
4,027
4,023
4,021
4,025
4,028
4,023
4,020
4,025
4,026
1043
1038
1031
1026
1079
1064
1047
1034
1186
1179
1173
1167
1219
1205
1190
1175
003
004
003
004
003
004
003
003
004
003
003
003
004
003
004
003
−005
−004
−005
−005
−002
−004
−004
−005
−004
−005
−005
−005
−004
−004
−004
−005
−05
−04
−05
−05
−02
−04
−04
−05
−03
−04
−04
−04
−03
−03
−03
−04
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
02
02
02
02
02
02
02
02
03
03
03
03
03
03
03
03
3
3
3
3
7
7
7
7
3
3
3
3
7
7
7
7
25
25
50
50
25
25
50
50
25
25
50
50
25
25
50
50
25
50
25
50
25
50
25
50
25
50
25
50
25
50
25
50
378
366
362
350
426
401
391
364
563
555
550
542
599
581
571
552
4,038
4,043
4,036
4,042
4,034
4,038
4,042
4,042
4,037
4,035
4,040
4,042
4,038
4,035
4,040
4,041
378
366
362
350
427
402
391
364
562
554
550
541
599
581
571
551
003
004
003
003
003
003
004
003
003
004
003
003
004
003
003
004
000
000
000
000
001
001
000
000
−001
−001
000
−001
000
000
000
−001
00
00
00
00
02
02
00
00
−02
−02
00
−02
00
00
00
−02
90
90
90
90
90
90
90
90
90
90
90
90
90
90
90
90
02
02
02
02
02
02
02
02
03
03
03
03
03
03
03
03
3
3
3
3
7
7
7
7
3
3
3
3
7
7
7
7
25
25
50
50
25
25
50
50
25
25
50
50
25
25
50
50
25
50
25
50
25
50
25
50
25
50
25
50
25
50
25
50
075
065
068
059
103
082
087
066
192
185
184
177
219
203
201
184
4,033
4,031
4,031
4,029
4,033
4,031
4,030
4,029
4,025
4,024
4,027
4,030
4,025
4,023
4,028
4,028
076
066
069
060
104
083
088
067
193
186
185
178
220
203
202
185
003
004
004
004
003
004
003
003
004
003
003
004
003
003
004
003
001
001
001
001
001
001
001
001
001
001
001
001
001
000
001
001
13
15
15
17
10
12
11
15
05
05
05
06
05
00
05
05
K
Note. The CPU times are in seconds.
compute one price, irrespective of the parameter
ranges, while the lattice method works much slower,
taking over one hour to compute one price.
4.
Pricing Other
Path-Dependent Options
Lookback and barrier options are among the most
popular path-dependent options traded in exchanges
worldwide, and in over-the-counter markets; perpetual American options are interesting because they
serve as simple examples to illustrate finance theory.10
We shall demonstrate in this section that in the double
10
Pricing of barrier, lookback, and perpetual American options
also arises quite often in other contexts. For example, Merton
(1974), Black and Cox (1976), and more recently Leland (1994),
Longstaff (1995), Longstaff and Schwartz (1995), among others,
use lookback and barrier options to value debt and contingent
Kou and Wang: Double Exponential Jump Diffusion Model
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Management Science 50(9), pp. 1178–1192, © 2004 INFORMS
Table 2
Comparison of the Approximation and the True Value for Finite-Horizon American Put Option with t = 1 Year
Parameter values
1
2
“True” value (a)
CPU time
Approx. (b)
CPU time
Abs. error (b) − (a)
Relative error
((b) − (a))/(a) (%)
110
110
110
110
110
110
110
110
110
110
110
110
110
110
110
110
02
02
02
02
02
02
02
02
03
03
03
03
03
03
03
03
3
3
3
3
7
7
7
7
3
3
3
3
7
7
7
7
25
25
50
50
25
25
50
50
25
25
50
50
25
25
50
50
25
50
25
50
25
50
25
50
25
50
25
50
25
50
25
50
1237
1217
1204
1184
1329
1285
1254
1208
1579
1563
1551
1536
1651
1617
1589
1553
4,026
4,026
4,025
4,025
4,026
4,026
4,024
4,024
4,025
4,027
4,028
4,028
4,025
4,028
4,028
4,028
1232
1211
1200
1178
1327
1279
1254
1203
1576
1559
1549
1532
1651
1614
1591
1552
004
003
004
003
003
004
003
003
004
004
003
003
004
003
003
004
−005
−006
−004
−006
−002
−006
000
−005
−003
−004
−002
−004
000
−003
002
−001
−04
−05
−03
−05
−02
−05
00
−04
−02
−03
−01
−03
00
−02
01
−01
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
02
02
02
02
02
02
02
02
03
03
03
03
03
03
03
03
3
3
3
3
7
7
7
7
3
3
3
3
7
7
7
7
25
25
50
50
25
25
50
50
25
25
50
50
25
25
50
50
25
50
25
50
25
50
25
50
25
50
25
50
25
50
25
50
660
636
626
601
757
707
683
628
1010
994
983
967
1081
1046
1022
985
4,027
4,025
4,025
4,025
4,026
4,026
4,024
4,025
4,025
4,027
4,028
4,029
4,025
4,026
4,036
4,028
662
637
629
603
762
709
688
631
1013
996
987
970
1086
1049
1029
989
003
003
004
003
003
004
003
004
003
003
004
003
003
003
004
002
002
001
003
002
005
002
005
003
003
002
004
003
005
003
007
004
03
02
05
03
07
03
07
05
03
02
04
03
05
03
07
04
90
90
90
90
90
90
90
90
90
90
90
90
90
90
90
90
02
02
02
02
02
02
02
02
03
03
03
03
03
03
03
03
3
3
3
3
7
7
7
7
3
3
3
3
7
7
7
7
25
25
50
50
25
25
50
50
25
25
50
50
25
25
50
50
25
50
25
50
25
50
25
50
25
50
25
50
25
50
25
50
291
270
266
246
368
324
312
266
579
565
558
543
642
609
592
559
4,025
4,025
4,025
4,025
4,026
4,025
4,024
4,025
4,024
4,025
4,028
4,029
4,027
4,026
4,029
4,031
296
275
272
251
375
329
320
272
585
570
564
549
649
615
600
565
004
003
004
003
003
004
003
003
004
003
004
003
004
003
003
004
005
005
006
005
007
005
008
006
006
005
006
006
007
006
008
006
17
19
23
20
19
15
26
23
10
09
11
11
11
10
14
11
K
exponential jump diffusion model the closed-form
solutions for these options can still be obtained.
claims in corporate finance with endogenous default; McDonald and Siegel (1986) use perpetual American options in studying real options. Within the classical geometric Brownian motion
framework, closed-form solutions for lookback, barrier, and perpetual American options are available, at least since McKean (1965),
Merton (1973), Goldman et al. (1979), and Conze and Viswanathan
(1991).
4.1. Lookback Options
Because the calculation for the lookback call option
follows just by symmetry, we will only provide the
result for the lookback put option, whose price is
given by
LPT = E∗ e−rT max M max St − ST 0≤t≤T
= E∗ e−rT max M max St
− S0
0≤t≤T
Kou and Wang: Double Exponential Jump Diffusion Model
1184
Management Science 50(9), pp. 1178–1192, © 2004 INFORMS
where M ≥ S0 is a fixed constant representing the
prefixed maximum at time 0.
Theorem 1. Using the notations ,1 ++r and ,2 ++r
as in (4), the Laplace transform of the lookback put is
given by
0
e−+T LPT dT
−Ke−rT
S0A+ S0 ,1 ++r −1 S0B+ S0 ,2 ++r −1
+
=
C+
M
C+
M
+
M
S0
−
++r
+
1 − ,1 ++r ,2 ++r
,1++r − 1
K
H
log
log
T
S0
S0
· 7 r − 12 2 − p 1 2 8
K
H
log
log
T (11)
S0
S0
where p̃ = p/1 + · 1 /1 − 1, ˜ 1 = 1 − 1, ˜ 2 =
2 + 1, ˜ = + 1, with given in (3) and 7 in (10).
for all + > 0; here
A+ =
Theorem 2. The price of the UIC option is obtained as
˜ p̃ ˜ 1 ˜ 2 8
UIC = S07 r + 12 2 − B+ =
,2 ++r − 1 ,1 ++r
,2 ++r − 1
C+ = + + r1 ,2 ++r − ,1 ++r The proof of Theorem 1 will be given in
Appendix B.1. Essentially, the proof explores a link
between the Laplace transform of the lookback option
and the Laplace transform of the first passage times
of the double exponential jump diffusion process as
solved explicitly in Kou and Wang (2003).
4.2. Barrier Options
There are eight types of (one dimensional, single)
barrier options: up (down)-and-in (out) call (put)
options. For example, the price of a down-and-out
put (DOP) option is given by DOP = E∗ e−rT K −
ST + 1min0≤t≤T St≥H !, where H < S0 is the barrier
level. Since all the eight types of barrier options can
be solved in similar ways, we shall only illustrate
with the up-and-in call (UIC) option, whose price is
given by
UIC = E∗ e−rT ST − K+ 1max0≤t≤T St≥H where H > S0 is the barrier level. Introduce the following notation: For any given probability P,
7 p 1 2 8 a b T = P ZT ≥ a max Zt ≥ b 0≤t≤T
(10)
where under P, Zt is a double exponential jump diffusion process with drift , volatility , and jump rate
N t
, i.e., Zt = t + W t + i=1 Yi , and Y has a double exponential distribution with density fY y ∼ p ·
1 e−1 y 1y≥0 +q · 2 ey2 1y<0 . The formula of the UIC
option will be written in terms of 7 . The Laplace
transforms of 7 is computed explicitly in Kou and
Wang (2003).
The proof of Theorem 2 will be given in
Appendix B.2. It uses a change of numeraire argument, which intuitively changes the unit of the money
from the money market account to the underlying
asset St, to reduce the computation of the expectation to the difference of two probabilities. For further
background of the change of numeraire argument for
jump diffusion processes, see, for example, Schroder
(1999).
4.3.
Numerical Results for Barrier and
Lookback Options
Since the solutions for barrier and lookback options
are given in terms of Laplace transforms, numerical
inversion of Laplace transforms becomes necessary.
To do this, we shall use the Gaver-Stehfest algoˆ
rithm.
−+x Given the Laplace transform function f + =
e f x dx of a function f x, the algorithm gener0
ates a sequence fn x such that fn x → f x, n → .
The algorithm11 converges very fast; as we will see,
it typically converges nicely even for n between 5
and 10. Kou and Wang (2003) report the details of
implementation.
As a numerical illustration, we calculate both the
lookback put option and the UIC barrier option in
Table 3. For the lookback put option the predetermined maximum is M = 110; for the UIC option the
barrier and the strike price are given by H = 120 and
K = 100, respectively. The expiration dates for both
options are the same: T = 1. The risk-free rate is r =
5%. The parameters used in the double exponential
jump diffusion are = 02, p = 03, 1/1 = 002, 1/2 =
11
The main advantages of the Gaver-Stehfest algorithm are: simplicity (a very short code will do the job), fast convergence, and
good stability (i.e., the final output is not sensitive to a small perturbation of initial input). The main disadvantage is that it needs
high accuracy computation, because it involves calculation of some
factorial terms; typically 30–80 digit accuracy is needed. However,
in many software packages (e.g., “Mathematica”) one can specify
arbitrary accuracy, and standard subroutines for high precision calculation in various programming languages (e.g., C++) are readily
available.
Kou and Wang: Double Exponential Jump Diffusion Model
1185
Management Science 50(9), pp. 1178–1192, © 2004 INFORMS
Table 3
The Prices of Lookback Put and UIC Options
Lookback put
n
1
2
3
4
5
6
7
8
9
10
Total CPU time
Brownian motion case
Up-and-in call
= 001
=3
= 001
=3
1720214
1655458
1614481
1595166
1587823
1585473
1584823
1584664
1584629
1584622
0541 sec
1584226
1858516
1783041
1735415
1713035
1704556
1701851
1701105
1700923
1700884
1700877
0711 sec
N.A.
1032186
981060
949290
934838
929672
928173
927813
927740
927726
927724
2849 min
927451
1098416
1062657
1031446
1013851
1007120
1005461
1005280
1005309
1005315
1005307
2815 min
N.A.
9.14
890
938
9.24
900
948
9.82
956
1008
10.05
(9.79, 10.31)
Monte Carlo simulation
200 points
CPU time: 8 min
2
000 points
CPU time: 37 min
point est.
95% C.I.
point est.
95% C.I.
15.39
1522
1556
15.65
1547
1583
16.29
1606
1652
16.78
1659
1697
Note. The Monte Carlo results are based on 16,000 simulation runs.
004, = 3, S0 = 100. To make a comparison with the
limiting geometric Brownian motion model = 0,
we also use = 001. The results are given in Table 3.
All the computations are done on a Pentium 700 PC.
The simulation results from the standard Monte
Carlo, along with the number of discretization points
used, are also reported in the table. Note that the
Monte Carlo simulation has two sources of errors: the
random sampling error, and the systematic discretization bias. It is quite possible to significantly reduce
the random sampling error here (thus the width of the
confidence intervals) by using some variance reduction techniques, such as control variates and importance sampling. However, it is difficult to reduce the
systematic discretization bias, resulting from approximating the maximum of a continuous time process
by the maximum of a discrete time process in simulation.12 For both the lookback put and the UIC,
it makes the calculation from the simulation biased
low. Even in the Brownian motion case, because of
the presence of boundary, this type of discretization
bias is significant, resulting in a surprisingly slow rate
of convergence13 in simulating the first passage time,
12
Based on “sharp large deviation” for diffusion models, Baldi et al.
(1999) suggest a simulation method for pricing of barrier and lookback options; it remains to be seen whether the method can be
effectively generalized to simulate models with jumps.
13
Asmussen et al. (1995) show that, theoretically, the discretization
error has an order 1/2, which is much slower than the order 1
convergence for simulation without the boundary; 16,000 points are
suggested in the paper for a Brownian motion with drift −1 and
volatility = 1 and time T = 8.
both theoretically and numerically. Here, the standard Monte Carlo is used mainly because we focus
on analytical solutions; for more efficent simulation
methods, see the recent paper by Metwally and Atiya
(2002), which proposes two schemes that reduce the
discretization bias in Monte Carlo pricing of barrier
options under jump diffusion models.
4.4. Perpetual American Options
To simplify the derivation, we shall only focus on the
perpetual American put option, because the methodology is also valid for the perpetual American call
option with dividends. Under the jump diffusion
model, the price of an American put option is given by
2S0 = sup# E∗ e−r# K − S#+ ! = sup# E∗ e−r# K−
S0eX# + !, where the supremum is taken over all
stopping times # taking values in 0 !.
Theorem 3. Using14 the notations ,3 r and ,4 r as
in (4), the value15 of the perpetual American put option is
given by 2S0 = V S0, where the value function V
is given by
K − v
if v < v0
V v =
(12)
−,3 r
−,4 r
Av
+ Bv
if v ≥ v0
14
15
Actually, ,1 r = 1.
Gerber and Shiu (1998) and Mordecki (1999) study the same optimal stopping problem with one-sided jumps (can only jump up
or down); this may not have the overshoot problem if the process
always jumps away from (not jump towards) the boundary. Also
r = 0 in Mordecki (1999). Here we focus on the (two-sided) double
exponential jump diffusion processes with r ≥ 0.
Kou and Wang: Double Exponential Jump Diffusion Model
1186
Figure 2
Management Science 50(9), pp. 1178–1192, © 2004 INFORMS
Values of American Put Options
Option Value
Option Value
20
18
16
14
12
10
13.5
13
12.5
12
11.5
90
100
110
120
s0
2
4
6
8
10
Lambda
Option Value
Option Value
13
12.2
12.5
30
40
50
60
70
80
Eta1
11.8
12
11.5
Eta2
11.6
30
Option Value
60
70
80
35
11.6
30
11.5
25
11.4
20
0.2
0.4
0.6
0.8
p
,3 r
,4 r
+1
·
·
v0 = K 2
2
1 + ,3 r 1 + ,4 r
,4 r
,3 r 1 + ,4 r
A = v0
K − v0 > 0
,4 r − ,3 r 1 + ,4 r
1 + ,3 r
,3 r
,
v0 −
K > 0
B = v0 4 r
,4 r − ,3 r
1 + ,3 r
Furthermore, the optimal stopping time is given by # ∗ =
inft ≥ 0 St ≤ v0 .
The proof16 will be given in Appendix B.3.
Note that the solution given in (12) satisfies the
16
50
Option Value
40
11.7
where
40
A result similar to (12) is also independently obtained by
Mordecki (2002). However, there are two key differences. First, our
proof not only covers the case of the perpetual American options,
but also solves another infinite-horizon free boundary problem
(with a more complicated boundary condition) of (A3) and (A4),
arising in approximating the finite-horizon American options; see
Appendix A.1. Second, the proof in Mordecki (2002) shows the
results indirectly, as it first derives some general representations
for Lévy processes, and then shows that the representations can be
computed explicitly if the jump sizes are exponentially distributed.
Here we prove and calculate the result directly by using martingale
and PDE methods, without appealing to more general results from
Lévy processes.
0.25
0.30 0.35
0.40 0.45
Sigma
0.50
smooth-fit principle (i.e., the value function is continuous and continuously differentiable across the free
boundary v0 ).17
Figure 2 graphs of the value of a perpetual
American put option versus its parameters, S0, 1 ,
2 , p, . The defaulting parameters are r = 006, =
020, K = 100, S0 = 100, = 3, p = 03, 1/1 = 002,
and 1/2 = 003. It only takes less than one second
to generate all the pictures in Figure 2 on a Pentium
700 PC. Not surprisingly, Figure 2 indicates that the
option value is a decreasing function of S0, p, and is
an increasing function of , 1/2 , and , as it is a put
option. What is interesting is that the option value is
an increasing function of 1/1 , which is the mean of
the positive jumps. The reason is that the risk-neutral
drift also depends on 1 ; a similar phenomenon was
also pointed out in Merton (1976).
5.
Concluding Remarks
Both the normal jump diffusion model and the double
exponential jump diffusion model are special cases
17
The smooth-fit principle may not hold for general Lévy processes;
see Pham (1997) and Boyarchenkov and Levendorskiı̆ (2002a),
where sufficient conditions for the smooth-fit principle are given.
Kou and Wang: Double Exponential Jump Diffusion Model
1187
Management Science 50(9), pp. 1178–1192, © 2004 INFORMS
of the affine jump diffusion models (Duffie et al.
2000, Chacko and Das 2002), which include stochastic
volatility and jumps in the volatility, and of Lévy processes, which have independent increments but with
more general distributions. Whether the double exponential jump diffusion model is suitable for modeling
purposes is ultimately a choice between analytical
tractability and reality, and it should be judged on
a case-by-case basis. (For example, the independent
increment assumption is perhaps more defensible in
the case of currency markets than in the case of equity
markets.) See Cont and Tankov (2002) for calibration
of the double exponential jump diffusion model to
market data, and some empirical comparison with
other models.
It is worth mentioning that jump diffusion models
may also be useful in modeling credit risks. In fact,
Huang and Huang (2003) have used the double exponential jump diffusion model to study credit spread,
and the empirical results there seem to be promising.
From a broader perspective, the paper calls for consideration of using exponential type distributions in
modeling jumps in asset pricing, and, by understanding the simplest cases first, the results may hopefully
shed some light on more general models with jumps.
Acknowledgments
Introduce the change of variable z = 1 − e−rt , gx z =
<x t/z. It is easy to see that zt = re−rt , <x = zgx , <xx = zgxx ,
<t = zt g + zgz zt . Plugging this back into (A1), and dividing
z on both sides, we have
r
+ g + 12 2 gxx + r − 12 2 − > gx
−r1 − zgz −
z
+
gx + y zfY y dy = 0
−
for all x > x0 t and gx z = 1/zK − ex − EuPex t,
∀x ≤ x0 t. Following Barone-Adesi and Whaley (1987), the
approximation will set
1 − zgz ≈ 0
(A2)
This is a reasonable assumption especially for very big or
very small t. Indeed, as t → 0, 1 − z → 0, while as t → ,
gz → 0, because gx z converges to the price of a perpetual American put option. This also explains why, in the
numerical tables, the error tends to be larger when t = 1.
With Approximation (A2), the function g satisfies the following equations:
r
−
+ g + 12 2 gxx + r − 12 2 − gx
z
(A3)
+
−
gx + y zfY y dy = 0
∀x > x0 t
and
The authors thank an anonymous referee, the associate editor, and the editor for many helpful comments. We are also
grateful to many people who offered insights into this work,
including seminar participants at various universities, and
conference participants at Risk Annual Boston Meeting
2000, UCLA Mathematical Finance Conference 2001, Taipei
International Quantitative Finance Conference 2001, Fifth
SIAM Conference on Control and Its Applications 2001,
American Finance Association Annual Meeting 2001, and
the 11th Derivatives Conference in New York City 2003. We
would like to thank Giovanni Petrella for his assistance in
computing the numerical values of American options. All
errors are ours alone.
If we regard t, and hence z and x0 t, to be fixed, (A3)
becomes an ordinary integro-differential equation with freeboundary x0 t. Note that the boundary condition in (A4)
involves the European put option price EuPex t, which
makes solving the free boundary problem more difficult
than that for perpetual American options. Under the
assumption of exponential jump size distribution, however,
the above free boundary problem can be solved explicitly
as in Appendix A.2, resulting in the approximation in (7).
Appendix A. Derivation of Approximation (7)
A.2.
A.1. Outline of the Main Steps in the Derivation
Let t be the remaining time to maturity. Suppose the optimal exercise boundary is v0 t; in other words, it is optimal
to exercise the option whenever the stock price falls below
v0 t. Letting x0 t = logv0 t, x = logv, and using the
generator in (5), the value function V x t = 2ex t and
the optimal exercise boundary v0 t must satisfy the free
boundary problem: −Vt − rV + V = 0, for x > x0 t; and
V x t = K − ex , for x ≤ x0 t. Define the early exercise premium as <x t = V x t − EuPex t. Since the European
put price satisfies the equation −EuPt − rEuP + EuP = 0,
for all x, it follows that the early exercise premium satisfies
the equation
−<t − r< + < = 0
x
∀x > x0 t8
x
<x t = K − e − EuPe t
∀x ≤ x0 t
(A1)
1
gx z = K − ex − EuPex t
z
∀x ≤ x0 t
(A4)
Solving the Free Boundary Problems (A3) and (A4)
Lemma 1. Define
x
x = K − e − hx
V
−x,3
Ae
+ Be−x,4 if x < x0
if x ≥ x0
where ,3 , ,4 > 0, x0 ∈ − are arbitrary constants, and hx
arbitrary continuous function. Then for any constant b, we have
for all x > x0 ,
+ V
x
−b V
= Ae−x,3 f˜,3 +Be−x,4 f˜,4 +q2 ex0 −x2
K
ex0
Ae−x0 ,3 Be−x0 ,4 0
·
−
−
−
−
hx0 +yey2 dy 2 1+2 2 −,3 2 −,4
−
where f˜x = G−x − b.
Kou and Wang: Double Exponential Jump Diffusion Model
1188
Management Science 50(9), pp. 1178–1192, © 2004 INFORMS
x + y dF y,
Proof. First, we want to compute − V
x. For
which is essential to compute the generator V
x > x0 , we have
x + y dF y
V
−
=
x0 −x
−
+
0
x0 −x
Ae−,3 y+x + Be−,4 y+x q2 ey2 dy
= E∗
q2 B −,4 x
e
− e−,4 x0 · ex0 −x2 !
2 − ,4
p1 e−,4 x
p1 e−,3 x
+B
+ A
1 + ,3
1 + ,4
+
= 12 2 A,23 e−x,3 + B,24 e−x,4 + r − 12 2 − −A,3 e−x,3 − B,4 e−x,4 − bAe−x,3 + Be−x,4 − Ae−x,3 + Be−x,4 2 ex0
x0 −x2
K−
+ qe
1 + 2
q2 A −,3 x
− e−,3 x0 · ex0 −x2
e
+
2 − ,3
0
hx0 + yey2 dy
− q2 ex0 −x2
−
q2 B −,4 x
− e−,4 x0 · ex0 −x2
e
+
2 − ,4
p1 e−,4 x
p1 e−,3 x
+B
+ A
1 + ,3
1 + ,4
= Ae−x,3 f˜,3 + Be−x,4 f˜,4 Ae−x0 ,3 2 Be−x0 ,4
ex0
− 2
−
+ qex0 −x2 K − 2
1 + 2
2 − ,3
2 − ,4
0
− 2
hx0 + yey2 dy −
from which the proof is terminated. Lemma 2. For every x0 , we have
A
EuPex t
= EuPex0 t−Ke−rt P∗ St ≤ K S0 = ex0 Ax
x=x0
0
EuPex0 +y te2 y dy
−
Ke−rt
1
EuPex0 t+
P∗ St ≤ K S0 = ex0 2 +1
2 2 +1
Ke−rt
St −2
+
·E∗
1St>K S0 = ex0 2 2 +1
K
ex
ex
K−ex eXt
−
e−rt 1y≥0 dy
e−rt eXt 1K−zeXt ≥0 dz
E∗ e−rt eXt 1K−zeXt ≥0 dz
A
EuPex t
= −ex0 · E∗ e−rt eXt 1K−ex0 eXt ≥0 Ax
x=x0
from which the first equation follows readily. As for the
second equation, we have
0
−
EuPex0 +y te2 y dy
0
−
= e−rt E∗
+ V
x
−b V
Hence
= E∗
Next, for x > x0 , we have
=
−
=
EuPex t = E∗ e−rt K − ex eXt + ! = E∗
K − ey+x − hx + yq2 ey2 dy
Ae−,3 y+x + Be−,4 y+x p1 e−y1 dy
0
2 ex0
q2 A −,3 x
x0 −x2
e
− e−,3 x0 · ex0 −x2 !
= qe
K−
+
1 + 2
2 − ,3
x0 −x
hx + yq2 ey2 dy
−
+
Proof. We have
e−rt K − ex0 +y eXt + · e2 y dy
0
−
1K−ex0 eXt ≥0 e2 y K − ex0 +y eXt dy
logK/eXt −x0
1K−ex0 eXt <0 e2 y K − ex0 +y eXt dy
+ e−rt E∗
−
St
K
1K−St≥0
−
= e−rt E∗
2 2 + 1
+
K 2 +1
· e−rt E∗ St−2 1K−St<0 2 2 + 1
from which the conclusion follows. Now we are in a position to solve the free boundary
problem of (A3) and (A4). Since <x t = zgx t, it is not
difficult to see that (A3)–(A4) reduce to −r/z< + < = 0,
∀x > x0 t; <x t = K − ex − EuPex t, ∀x ≤ x0 t. Note we
regard t as fixed. Denote x0 = x0 t. By Lemma 1, for < =
Ae−,3 x + Be−,4 x with x ≥ x0 , we must have G−,3 − r/z = 0,
G−,4 − r/z = 0, and
0
K
ex0
−
−
EuPex0 +y t · e2 y dy
2 1 + 2
−
=
Be−,4 x0
Ae−,3 x0
+
2 − ,3 2 − ,4
(A5)
The smooth-fit principle (i.e., the value function is continuous and continuously differentiable across the free boundary) yields two more equations K − ex0 − EuPex0 t =
Ae−,3 x0 + Be−,4 x0 , ex0 + A/AxEuPex tx=x0 = A,3 e−,3 x0 +
B,4 e−,4 x0 These five equations determine the five unknown
parameters A, B, x0 , ,3 , and ,4 . It is not difficult to verify
that A and B are given by
1
, K − ex0 − EuPex0 t
A = e,3 x0 ·
,4 − ,3 4
A
− ex0 + EuPex t
Ax
x=x0
1
, K − ex0 − EuPex0 t
B = e,4 x0 ·
,3 − ,4 3
A
− ex0 + EuPex t
Ax
x=x0
Kou and Wang: Double Exponential Jump Diffusion Model
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Management Science 50(9), pp. 1178–1192, © 2004 INFORMS
After some algebra, (A5) yields that the free boundary x0
must satisfy the equation:
,3 ,4
, + ,4 + 1 + ,4 ,3
K − ex0 3
2
1 + 2
A
= EuPex t
+,3 + ,4 − 2 EuPex0 t
Ax
x=x0
0
+ 2 − ,4 2 − ,3 EuPex0 +y t · e2 y dy
−
Using Lemma 2, we have
A=
e,3 x0 , K − 1 + ,4 ex0 + EuPex0 t!
,4 − ,3 4
+ Ke−rt P∗ St ≤ K S0 = ex0 ! e,4 x0 , K − 1 + ,3 ex0 + EuPex0 t!
B=
,3 − ,4 3
+ Ke−rt P∗ St ≤ K S0 = ex0 ! which are exactly (8) and (9). Again using Lemma 2, we
have that x0 must satisfy
,3 ,4
1 + ,3 1 + ,4 K − ex0
2
1 + 2
1 + ,4 1 + ,3 = EuPe t
2 + 1
x0
+
,4 ,3 − 2 − 2 ,3 − ,4 2 −rt ∗
Ke P St ≤ K S0 = ex0 !
2 2 + 1
+ 2 − ,4 2 − ,3 Ke−rt
St −2
·
E∗
ISt ≥ K S0 = ex0 2 2 + 1
K
Since 2 is typically very large, as 1/2 is about 2% to
10%, the last term in the above equation is generally very
small, because the expectation E∗ St/K−2 ISt ≥ K S0 = ex0 ! is typically small. Ignoring the last term, we have
that x0 must satisfy the equation
,3 ,4
1+,3 1+,4 K −ex0
2
1+2
= EuPex0 t·
+
1+,4 1+,3 2 +1
−2 −2 ,3 −,4 2 +,4 ,3 −rt ∗
Ke ·P St ≤ K S0 = v0 !
2 2 +1
which is exactly (6).
It remains to show that A > 0 and B > 0. To do this, we
need the following lemma.
Lemma 3. For the unique solution v0 in Equation (6), we
have
K > v0 + EuPv0 t = e−rt E∗ maxSt K S0 = v0 !
(A6)
Proof. We show this by contradiction. First, note that
v0 + EuPv0 t = e−rt Ev0 maxSt K! is an increasing function of v0 . Next, assume by contradiction that K ≤ v0 +
EuPv0 t. Since C, − D, = ,4 ,3 − 2 1 + ,3 + ,4 < 0,
we have
C, K − D, v0 + EuPv0 t!
≤ C, − D, K < C, − D, Ke−rt · Pv0 St ≤ K!
Thus, the left side of (6) would be smaller than the right
side of (6), a contradiction. Now we can show that A B > 0. By taking derivative and
then using (A6), it is easy to see that the function
C, K − D, v0 + EuPv0 t!
− C, − D, Ke−rt · Pv0 St ≤ K!
(A7)
is strictly increasing in ,3 and strictly decreasing in ,4 .
Replacing ,3 by 2 in (A7) and observing ,3 < 2 and (6),
we have ,4 K − 1 + ,4 v0 + EuPv0 t! + Ke−rt · Pv0 St ≤ K!
> 0, yielding A > 0. Similarly, since ,4 > 2 , replacing ,4 by
2 in (A7) yields B > 0. Appendix B. Proofs for Other Path-Dependent
Options
B.1.
Proof of Theorem 1 for Lookback Options
Lemma 4. We have limy→ ey P∗ MX T ≥ y! = 0, ∀T ≥ 0.
Proof. It is not difficult to see that the process
e&Xt−G&t 8 t ≥ 0 is a martingale for any & ∈ −2 1 . Fix
an & ∈ 1 1 such that G& > 0 (such & always exists since
G1 = r ≥ 0). It follows that ey P∗ MX T ≥ y! = e1−&y ·
e&y P∗ MX T ≥ y! = e1−&y · e&y P∗ #y ≤ T !, where #y is the
first passage time of process X over level y; however, the
second term in the previous equation is dominated by
e&y P∗ #y ≤ T ! ≤ E∗ e&X#y ∧T !
≤ eG&T E∗ e&X#y ∧T −G&·#y ∧T ! = eG&T where the last equality follows from the optional sampling
theorem. Since & > 1, the result follows readily. Now we are ready to prove Theorem 1. Note that we only
need to compute the Laplace transform of Ls M8 T =
E∗ e−rT maxM seMX T !, M ≥ s, where s = S0 and M
are constants, and MX T = max0≤t≤T Xt. Letting z =
logM/s ≥ 0, we have
Ls M8 T = s E∗ e−rT maxez eMX T !
= s E∗ e−rT eMX T − ez 1MX T ≥z + sez e−rT Integration by parts yields
E∗ e−rT eMX T 1MX T ≥z
ey d P∗ MX T ≥ y!
= −e−rT
z
= −e−rT −ez P∗ MX T ≥ z! −
∗
=E e
−rT z
e 1MX T ≥z + e
−rT
z
z
P∗ MX T ≥ y!ey dy
ey P∗ MX T ≥ y! dy8
Kou and Wang: Double Exponential Jump Diffusion Model
1190
Management Science 50(9), pp. 1178–1192, © 2004 INFORMS
here we
have used Lemma 4. It follows that Ls M8 T =
se−rT z ey P∗ MX T ≥ y! dy + Me−rT . Therefore, for any
+ > 0,
e−+T Ls M8 T dT
0
Note that this is a well-defined probability as E∗ e−rt St/
S0 = 1. We have, by the Girsanov theorem for jump pro t = W t − t is a new Brownian motion under
cesses, W
P, and the original process
N t
Xt = r − 12 2 − t + W t +
Yi
M
e−+T e−rT
ey P∗ MX T ≥ y! dy dT +
=s
+
+r
0
z
M
ey
e−++rT P∗ MX T ≥ y! dT dy +
=s
++r
z
0
However, it follows from Kou and Wang (2003) that
e−++rT P∗ MX T ≥ y! dT = A1 e−y,1 ++r + B1 e−y,2 ++r 0
A1 =
,2 ++r
1 1 − ,1 ++r
·
++r
1
,2 ++r − ,1 ++r
B1 =
,1 ++r
1 ,2 ++r − 1
·
++r
1
,2 ++r − ,1 ++r
i=1
N t
t +
= r + 12 2 − t + W
Yi
i=1
is a new double exponential jump diffusion process with
the Poisson process N t having a new rate ˜ = E∗ eY =
1 + , and the jump sizes Y ’s being i.i.d. with a new
density given by
1
ey f y
E∗ eY Y
=
Note that ,2 ++r > 1 > 1, ,1 ++r > ,1 r = 1. Therefore,
e−+T Ls M8 T dT
0
=s
z
= sA1
=s
ey A1 e−y,1 ++r dy + s
−z,1 ++r −1
z
ey B1 e−y,2 ++r dy +
M
++r
−z,2 ++r −1
M
e
e
+ sB1
+
,1 ++r − 1
,2 ++r − 1
++r
A+ −z,1 ++r −1
B
M
e
+ s + e−z,2 ++r −1 +
C+
C+
++r
This yields the Laplace transform we obtained in
Theorem 1. B.2. Proof of Theorem 2 for Barrier Options
We can write UIC as
UIC = E∗ e−rT ST − K+ 1max0≤t≤T St≥H = E∗ e−rT ST 1ST ≥K max0≤t≤T St≥H − Ke−rT P∗ ST ≥ K max St ≥ H
=p
It is easy to compute the second term, as
II = 7 r − 12 2 − p 1 2 8
log
H
K
log
T S0
S0
For the first term, we can use a change of numeraire
argument. More precisely, introduce a new probability P
defined as
d
P ST = e−rT eXT = e−rT
dP∗ t=T
S0
N T 1 2
= exp − 2 − T + W T +
Yi i=1
ey p1 e−1 y 1y≥0 +
1
E∗ eY ey q2 e2 y 1y<0
1
1
·
− 1e−1 −1y 1y≥0
E∗ eY 1 − 1 1
+q
1
2
·
+ 1e2 +1y 1y<0 E∗ eY 2 + 1 2
Thus, it is still a double exponential density with ˜ 1 = 1 −1,
˜ 2 = 2 + 1,
q2 −1 1
p1
+
p̃ = p
1 − 1 2 + 1
1 − 1
q2 −1 2
p1
+
q̃ = q
1 − 1 2 + 1
2 + 1
In summary, we have
∗
−rT ST ·1
I = S0E e
S0 ST ≥K max0≤t≤T St≥H
= S0
P ST ≥ K min St ≤ H
0≤t≤T
˜ p̃ ˜ 1 ˜ 2 8
= S07 r + 12 2 − log
0≤t≤T
= I − Ke−rT · II (say)
1
E∗ eY H
K
log
T S0
S0
and UIC = I − Ke−rT · II, from which the conclusion
follows. B.3.
Proof of Theorem 3 for Perpetual
American Options
Lemma 5. Suppose there exist some x0 < log K and a nonnegative C 1 function ux such that: (1) the function u is C 2
on \x0 , and is convex with u x0 − and u x0 + existing;
(2) ux − rux = 0 for all x > x0 ; (3) ux − rux < 0
for all x < x0 ; (4) ux > K − ex + for all x > x0 ; (5) ux =
K − ex + for all x ≤ x0 ; and (6) there exists a random variable Z
∗
with E∗ Z! < , such that e−rt∧#∧# uXt ∧ # ∧ # ∗ + x ≤ Z,
for any t ≥ 0, x and any stopping time #. Here the infinitesimal
generator is defined in (5). Then the option price 2S0 =
ulogS0 and the optimal stopping time is given by # ∗ =
inft ≥ 0 St ≤ ex0 .
Kou and Wang: Double Exponential Jump Diffusion Model
1191
Management Science 50(9), pp. 1178–1192, © 2004 INFORMS
= x + Xt. Then Xt
Proof. Define Xt
has the same
generator . The result now follows from a similar argument in Mordecki (1999, pp. 230–232), except with Mt
t
being changed to Mt = e−rt uXt
− 0 e−rs −ruXs
+
uXs ds. Proof of Theorem 3. Let x = logv and x0 = logv0 .
Then
K − ex if x < x0
V x =
Ae−x,3 r + Be−x,4 r if x ≥ x0 For notation simplicity, we shall write ,3 ≡ ,3 r and ,4 ≡
,4 r . To prove Theorem 3, we only need to check the conditions in Lemma 5. Note that f ,3 = f ,4 = 0, and
x0
K − e = Ae
−x0 ,3
+ Be
0=K−
x0
−x0 ,4
x0
e = A,3 e
−x0 ,3
−x0 ,3
+ B,4 e
−x0 ,4
=
0
−
+
x0 −x
0
x0 −x
K − ey+x p1 e−y1 dy
Ae−,3 y+x + Be−,4 y+x p1 e−y1 dy
p1
q2
+
2 + 1 1 − 1
e−x0 ,3
e−x0 ,4
ex0
−A 1
−B 1
− pe−1 x0 −x K − 1
1 − 1
1 + ,3
1 + ,4
= K − ex
Therefore, for x < x0 ,
−rV + V x
= − 12 2 ex + r − 12 2 − −ex − rK − ex − K − ex p1
q2
x
+
− pe−1 x0 −x
+ K −e
2 + 1 1 − 1
ex0
e−,4 x0
e−,3 x0
· K− 1
−B 1
−A 1
1 − 1
1 + ,3
1 + ,4
Rearranging terms and using (3) we have for x < x0 ,
−rV + V x = −rK − e−1 x0 −x p
e−,3 x0
e−,4 x0
ex0
−A 1
−B 1
· K− 1
1 − 1
1 + ,3
1 + ,4
The right-hand side can be further simplified as
Ae−,3 x0 1 Be−,4 x0
1 ex0
−
− 1
1 + ,3
1 + ,4
1 − 1
,4
1 + ,4
1
1 v0
K − v0
−
=K−
1 − 1 1 + ,3 ,4 − ,3 1 + ,4
,3
1 + ,3
1
K
v −
−
1 + ,4 ,4 − ,3 0 1 + ,3
K−
=K
1 + ,3 1 + ,4 ,3 ,4
− v0 1
1 + ,3 1 + ,4 1 − 11 + ,3 1 + ,4 = −K
= −rK + pe−1 x0 −x K
2 + 1
,3 ,4
1 + ,3 1 + ,4 2 1 − 1
,3 ,4 2 + 1 1 + ,3 1 + ,4 2 1 − 1
from which it is easy to see that −rV +V x is an increasing function, thanks to the assumption 1 > 1. Thus, to show
Condition 3 it suffices to show that −rV + V x0 − ≤ 0.
However, since V x is bounded and continuous, it follows
from the Dominated Convergence Theorem that
V x0 − − V x0 +
e 2
A2 e
B2 e
−
−
1 + 2
2 − ,3
2 − ,4
K − ey+x q2 ey2 dy +
−rV + V x
= 12 V x0 − − V x0 +
−x0 ,4
Therefore, Condition 2 follows from Lemma 1 with h = 0.
Conditions 1, 4, 5, and 6 are trivial by noting that V x0 + =
V x0 − and 0 ≤ V x ≤ K.
As to Condition 3, note that for x < x0 , we have
V x + ya dF y
−
In summary we have for x < x0 ,
= − 12 ex0 + ,23 Ae−,3 x0 + ,24 Be−,2 x0 ≤ 0
But −rV + V x0 + = 0, which completes the proof. References
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